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Numerical investigation of heat and mass transfer from an evaporating meniscus in a heated open groove
International Journal of Heat and Mass Transfer 54 (2011) 3015–3023
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical investigation of heat and mass transfer from an evaporating meniscus in a heated open groove Hao Wang a,⇑, Zhenhai Pan a, Suresh V. Garimella b a b
Energy and Resources Engineering Department, College of Engineering, Peking University, Beijing 100871, China Cooling Technologies Research Center, An NSF IUCRC, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA
a r t i c l e
i n f o
Article history: Received 30 July 2010 Received in revised form 29 January 2011 Accepted 18 February 2011 Available online 22 March 2011 Keywords: Evaporation Vapor transport Diffusion Meniscus Thin film Marangoni Groove Temperature valley
a b s t r a c t The process of evaporation from a meniscus into air is more complicated than in enclosed chambers filled with pure vapor. The vapor pressure at the liquid–gas interface depends on both of the evaporation and the vapor transport in the gas environment. Heat and mass transport from an evaporating meniscus in an open heated V-groove is numerically investigated and the results are compared to experiments. The evaporation is coupled to the vapor transport in the gas domain. Conjugate heat transfer is considered in the solid walls, and the liquid and gas domains. The flow induced in the liquid due to Marangoni effects, as well as natural convection in the gas due to thermal expansivity and vapor concentration gradients are simulated. The calculated evaporation rates are found to agree reasonably well with experimentally measured values. The convection in the gas domain has a significant influence on the overall heat transfer and the wall temperature distribution. The evaporation rate near the contact lines on either end of the meniscus is high. Heat transfer through the thin liquid film near the heated wall is found to be very efficient. A small temperature valley is obtained at the contact line which is consistent with the experimental observation. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction A variety of industrial applications have exploited the high heat transfer rates that can be supported by the phase change occurring in evaporating liquid menisci in capillary structures. A detailed understanding of the heat and mass transfer occurring in evaporation from menisci under different operating conditions is critical to the proper design of a variety of passive phase-change devices. A V-shaped groove is a type of capillary structure that has been widely studied. In many of the past studies, a closed system was considered in which the vapor domain was assumed to be uniform at saturated conditions, including in the region near the meniscus. Xu and Carey [1] developed an approximate model for meniscus evaporation in a V-groove. They assumed that the liquid flow along the groove was driven primarily by the capillary pressure difference due to the recession of the meniscus towards the apex of the groove. Khrustalev and Faghri [2] presented a mathematical model for the heat transfer through thin liquid films in the evaporator section of heat pipes with capillary grooves. The model accounted for the effects of interfacial thermal resistance, disjoining pressure, and surface roughness for a given meniscus contact angle. An analytical investigation of the heat transfer characteris⇑ Corresponding author. Tel.: +86 10 82529060. E-mail address: [email protected] (H. Wang). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.02.047
tics of evaporating thin liquid films in V-shaped microgrooves was conducted by Ha and Peterson [3]. The extended meniscus was divided into three regions: absorbed layer or non-evaporating region, evaporating thin film or transition region, and intrinsic meniscus region. Stephan and Busse [4] calculated the heat and mass transfer in the micro-region and then combined the solution with a treatment of the macroscopic meniscus within open grooves. Burelbach et al. [5] conducted a theoretical analysis of evaporating/condensing liquid films, in which vapor recoil and Marangoni convection were included. Schmidt [6] discussed the influence of Marangoni and buoyancy effects on the flow field near an evaporating meniscus. A number of studies [7–10] have also investigated the disjoining pressure due to long-range molecular forces as well as evaporation from the thin liquid film at the contact line. The transport process is more complicated, however, if the gas domain is not uniformly saturated with vapor but contains noncondensable gases as well. Although evaporation into unsaturated vapor domains is encountered widely in applications, it is less well-understood. Deegan et al. [11] investigated the drying process of water droplets in open space, showing that the evaporation from droplets was influenced by diffusion of vapor in the air. Cachile et al. [12] also investigated droplet drying in air and explained their experimental results by considering vapor transport in air. Dhavaleswarapu et al. [13] used micro-particle image velocimetry
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Nomenclature A C D Gr J hfg k l m mnet m00net M M n p R R Re T t V
area (m2) vapor molar concentration (mol/m3) diffusion coefficient in air (m2/s) Grashof number molar flux (mol /(m2 s)) latent heat of evaporation (J/kg) thermal conductivity (W/(m K)) characteristic length (m) mass (kg) mass flow rate (kg/s) mass flux (kg/(m2 s)) molecular weight (kg/mol) molecular weight (kg/kmol) interface normal coordinate (m) pressure (N/m2) radius (m) universal gas constant (J/(mol K)) Reynolds number temperature (K) time (s) fluid velocity (m/s)
r s
thickness (m) contact angle kinematic viscosity (m2/s) dynamic viscosity ((N s)/m2) density (kg/m3) surface tension coefficient (N/m) interface accommodation coefficient shear stress (N/m2)
Subscript diff e equ g l lv out ref sat v
diffusion evaporation equilibrium gas (vapor/air mixture) liquid interface outlet reference saturated vapor
d h
m l q r
Greek symbols b thermal expansion coefficient (1/K)
to reconstruct velocity fields in sessile water droplets evaporating into air. The spatial and temporal distribution of the local evaporative mass flux was determined, and the temperature distribution in the droplet near the contact line was estimated. Buffone et al. [14,15] conducted experiments on a meniscus in a microtube which was open to air. Similar experiments in [16] delineated Marangoni convection in capillary tubes. More recently, Pan and Wang [17] numerically investigated the broken symmetry of the Marangoni convections in micro-tubes, where a non-uniform vapor diffusion could be with responsibility. Migliaccio et al. [18] recently conducted detailed experiments for an evaporating heptane meniscus in a V-groove open to air. The groove was made of fused quartz, and electrical heating was provided by a thin layer of titanium coated on the backside of the quartz substrate. The effects of liquid feeding rate on the temperature suppression and meniscus shape were explored. Highresolution infrared thermography was employed to investigate the temperature profile on the groove wall. A small temperature valley was recorded at the contact line, indicating the localized cooling due to thin film heat transfer. The present work reports on a numerical investigation conducted for the evaporating meniscus in the open V-groove under conditions matching those of the experiments in [18]. The evaporation and vapor transport are coupled at the meniscus in the numerical model. The evaporation process, cooling effect and conduction in the groove wall, Marangoni flow, and convection in the vapor are all comprehensively investigated. The predicted results are compared to those from the experiments. 2. Mathematical model 2.1. Problem description The experiments in [18] considered a V-groove setup consisting of an angled heated quartz wall and a vertical center wall as shown in Fig. 1, simulating one half of a V-groove. The outer surface of the quartz wall is heated by means of electrical power supplied to a thin metallic film deposited on this surface. A meniscus is
sustained between the two walls, and its shape is determined by optical means. The half V-groove in the experiments is 50 mm long in the axial (z-) direction. To replace the liquid lost to evaporation, a syringe pump continuously feeds room-temperature liquid to the groove through a capillary tube positioned just above the meniscus contact line, 12.5 mm from one end of the groove. For the present modeling work, the liquid meniscus profile obtained at a liquid supply rate of 6.55 107 kg/s and an applied heat flux of 1800 W/m2 is chosen for investigation. A two-dimensional representation of the problem is adopted as shown in Fig. 1, for computational tractability. The mass loss due to evaporation in the liquid is not considered in the model since there is no liquid supply inlet in the 2D representation; this is a reasonable simulation of the experiment since the liquid consumption in the experiment is of order of 1 105 kg/(s m), with corresponding induced liquid flow velocities of less than 1 105 m/s. 2.2. Evaporation and vapor transport at the meniscus The evaporation from a liquid–gas meniscus is determined by several factors. The interfacial temperature Tlv is important since a higher Tlv brings more higher-energy liquid molecules to cross the interface into the vapor phase. The vapor pressure near the interface pv_lv is also an important factor because a higher pv_lv means that more vapor molecules could cross the interface back into the liquid phase, thus reducing the net evaporation mass flux m00net ; evaporation into a vacuum is more intensive than that into a vapor domain. When evaporating into air, the vapor molecules must diffuse through the air under a driving force. Thus, the vapor pressure near the interface pv_lv must be higher than that far from the interface. To be complete, analysis of evaporation at the meniscus must account for vapor transport in the gas domain. The net evaporated mass flux across the interface may be calculated according to the molecular kinetics-based evaporation theory of Schrage [19]:
M00net
2r M ¼ 2 r 2pR
!1=2
pv
equ ðT lv Þ 1=2 T lv
p v1=2lv T v lv
! ð1Þ
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With appropriate boundary conditions, the distribution of vapor concentration in the gas domain can be computed, and the mass flux due to vapor transport at the liquid–gas interface can be obtained as
Gravity Groove wall: quartz
0.004 0.003
Left side Heated underside
0.002
y (m)
m00v ¼ M J ¼ M ðD
0.001
Vapor / Air
0.000
Liquid
Meniscus in experiment
v n j lv ¼ Sealant junction
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007
x (m)
m00v ¼
C total ¼ 0.060
Boundary
MD @C v 1 C v =C total @n lv
ð7Þ
p RT lv
ð8Þ
and the vapor mole concentration Cv at the interface is determined from pv_lv under the ideal gas assumption:
Vapor / Air
0.040
p C v jlv ¼ v lv C total p
0.020
The liquid mass flux evaporated from the interface must be equal to the vapor mass flux at the interface by mass conservation, i.e., m00net ¼ m00v , which couples the evaporation rate to vapor transport at the interface. Combining Eqs. (3) and (7),
0.000
ð9Þ
Impermeable wall
pv lv ¼ pv
-0.020 -0.05
0
in which pv_equ(Tlv) is the equilibrium vapor pressure at which there would be no net mass flux across the interface. It is approximated as the corresponding saturation pressure, despite the implicit omission of suppression effects due to the disjoining pressure and the capillary pressure [20] in this approximation. The saturation pressure psat(Tlv) is calculated by the well-known Clausius–Clapeyron equation: equ ðT lv Þ psat ðT lv Þ ¼ psat
ref
exp
Mhfg R
1
T sat
ref
1 T lv
! ð2Þ
Assuming that the vapor temperature at the interface Tv_lv is equal to the liquid temperature at the interface Tlv,
m00net ¼
2r M 2 r 2pR
!1=2
1 T l1=2 v
ðpv
equ ðT lv Þ
pv lv Þ
ð3Þ
When the evaporation occurs into a uniformly vapor-saturated domain, pv_lv in Eq. (3) is known, and the evaporation mass flux m00net is obtained. In the present work, where the evaporation takes place in an air domain, pv_lv is unknown but may be calculated from a solution of the vapor transport problem. The governing equation for vapor transport in air, including the diffusion and convection components, is given as
@C v v rC v ¼ r ðD rC v Þ þ~ @t
equ ðT lv Þ
0.05
x (m)
Fig. 1. Schematic diagram of the open V-groove with an evaporating meniscus in the experiments [18].
pv
ð6Þ
where the total gas mole concentration Ctotal is
0.080
y (m)
1 @C air 1 @C v D ¼ D C air C total C v @n lv @n lv
Substituting this expression for vn into Eq. (5), the mass flux at the interface due to vapor transport is
Central wall: glass
-0.003 -0.001
ð5Þ
The first term on the right is the diffusion component while the second is the convection component; the use of vn ensures that the net mass transport of air is zero since air does not cross the interface, based on which the following expression may be written:
-0.001 -0.002
@C v þ v n C v Þ @n lv
ð4Þ
2r 2r
m00v 1=2
M 2pR
ð10Þ 1 1=2 T lv
In the computations, the vapor distribution Cv in the gas domain is first calculated with the initial boundary conditions. The vapor mass flux at the interface m00v is then obtained from Eq. (7). Eq. (10) is then used to calculate pv_lv, after which Cv at the interface is updated for the next round of vapor transport calculations. 2.3. Gas and liquid domains As explained above, vapor transport in the gas domain strongly influences evaporation at the meniscus. Careful calculation of the flow, temperature, and vapor concentration fields is therefore necessary. In the present work, the flow in the gas domain is assumed to be laminar and Newtonian, and the ideal-gas law is assumed to hold. With g = 9.8 m/s2, m = 1.6 105 m2/s, b = 0.0031 K1, and approximating l 0.01 m and DT 40 K, a value of the Grashof number (Gr = gl3bDT/m2) in the gas domain of Gr 4.7 103 is estimated, which justifies the assumption of laminar flow [21]. The density of the gas–vapor mixture (heptane vapor is heavier than air) is given by:
q ¼ C v Mv þ
p TR
C v M air
ð11Þ
and is not only a function of the local temperature T, but also the local concentration Cv. The continuity, momentum and energy equations must be solved together with the vapor transport equations (Eqs. (4) and (11)) to provide a complete description of convection in the gas domain. Boundary conditions on Cv are set as follows: the value of Cv at the interface is determined by Eq. (10), while the value at the outer boundary of the gas domain (a circle which is 0.1 m in diameter as
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shown in Fig. 1, 14 times the groove length) is assumed to be zero. The adequacy of the boundary size for the simulation is verified by considering a domain diameter that is half the selected size. In both cases, the evaporation mass flow rate was found to be 1.58 105 kg/(s m). The far-field boundary condition (Cv = 0) is also verified by running the simulation with a larger value of
Cv = 0.2 mol/m3. This only changes the evaporation mass flow rate very slightly to 1.57 105 kg/(s m). Experimental measurements of Cv in the V-groove domain would allow for greater fidelity in the numerical model. The flow in the liquid domain is assumed to be laminar, incompressible and Newtonian. The Boussinesq approximation is employed for modeling natural convection in the liquid, whereby the density changes with temperature only in the buoyancy term. Marangoni effects are also included. Mass transport across the interface is implemented in the layer of mesh cells adjacent to the interface on either side, as explained in [22–24]. The increase in mass in the gas due to evaporation is taken into account. A negative energy source term is employed on the liquid side to simulate the cooling effect due to evaporation. 2.4. Numerical analysis The numerical solution is obtained using the pressure-based finite volume scheme described in [25,26], and implemented in the commercial software package, FLUENT [27]. Pressure–velocity coupling is accomplished through the SIMPLE algorithm. The outer boundary is designated as a pressure-outlet boundary. The domain is discretized into quadrilateral elements as shown in Fig. 2. The very thin liquid film near the contact line is truncated in the mesh so that a film thickness less than 1 lm is neglected. 3. Results and discussion Results from the numerical model are presented for heptane evaporating in a V-groove made of quartz for the heated wall and glass for the center wall, to simulate the experiments in [18]. The parameters used in the simulation are listed in Tables 1 and 2. The predicted results are compared to the experimental measurements from [18]. 3.1. Heat transfer analysis In the experiment [18] the heat flux imposed on the quartz wall was 1800 W/m2. The wall width and length were 7.1 mm and 50 mm, respectively. The total heat imposed on the groove was therefore 0.64 W. For the present 2D model, this translates to a heat input of 12.78 W/m to simulate experimental conditions. Table 1 Fluid properties [28]. Thermophysical properties
Heptane
Air
Density (kg/m3)
666.95 at 294.65 K
Thermal conductivity (W/m K) Heat capacity (J/kg) Viscosity (kg/m s) Vapor molecular weight Thermal expansion coefficient dr/dT (N/m K) (313 K to 323 K) Vapor diffusion coefficient in air (m2/s) Accommodation coefficient Latent heat of evaporation (J/kg) Saturated vapor pressure (Pa)
0.12746 2298.4 0.000331 100 0.00128 0.000103 7.2 106 at 294.65 K 1 3.506 105 5402 at 294.65 K
Defined in Eq. (11) 0.0242 1006.43 1.789 105 29 – – – – – –
Table 2 Solid properties [18].
Fig. 2. Mesh setup for the V-groove system: (a) the full mesh, (b) details of the mesh in the vicinity of the groove, and (c) mesh near the contact line (the liquid film thinner than 1 lm is neglected).
Thermophysical properties
Fused quartz
Glass
RTV sealant
Density (kg/m3) Thermal conductivity (W/m K) Heat capacity (J/kg)
2200 1.4 670
2500 1.1 840
1500 0.2 920
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a. The evaporation heat transfer, Qe, is calculated from the model to be 5.54 W/m, accounting for 43.3% of the total heat input. The corresponding evaporation mass flow rate is 1.58 105 kg/(s m). This value is comparable to the liquid supply rate in the experiments of approximately 1.31 105 kg/(s m). b. The convection heat transfer from the unwetted portions of the groove wall, Qc1, contributes 4.66 W/m, accounting for 36.5% of the total heat input. This includes convection from the underside (2.53 W/m), the unwetted portion of the top side (1.62 W/m), the top edge (0.34 W/m), and the sealant area (0.17 W/m). c. The convection heat transfer from the vertical center wall, Qc2, is 1.28 W/m, which is 10% of the total input. d. In the experiments, the evaporating liquid is at the local temperature of the interface, while the liquid supply to the groove is at the room temperature of approximately 294.65 K. Therefore some of the heat input is absorbed by the supplied liquid to heat up to the interfacial temperature. This component Qs is directly calculated based on the simulated evaporation rate and the difference between room temperature and interface temperature to be 0.96 W/m, which is 7.5% of the total input. Since the liquid mass loss at the interface is not simulated in the model, this component of heat transfer is accounted for by subtracting it from the total heat input, effectively decreasing the applied heat flux at the heated boundary to 1665 W/m2 from 1800 W/m2. e. A final component Qc3 is dissipated from the meniscus by convection rather than by evaporation, and amounts to 0.35 W/m; this accounts for 2.7% of the total heat input, and is thus negligible. Of the total heat transfer, Qe + Qc1 + Qc2 + Qc3 + Qs = 12.78 W/m, phase change accounts for 43.3% while the convective components account for more than 50% of the total heat input.
value of 319 K is observed at the contact line with the vertical center wall. The local evaporation mass flux along the meniscus is plotted in Fig. 5. Near the contact line at the heated wall, the strongest evaporation flux is observed with a maximum of approximately 0.43 kg/(s m2), corresponding to a local heat flux of 1.3 105 W/ m2. This strong evaporation is due to the high local interfacial temperature. While the evaporation rate drops rapidly with distance away from the heated-wall contact line, there is a small increase as the contact line with the vertical center wall is approached. While this region has the lowest interfacial temperature, the improvement in evaporation rate is likely caused by the ease of vapor dissipation in this region, as can be surmised from the Cv contours shown in Fig. 6. The advantageous geometry in the region near the center-wall contact line facilitates vapor diffusion from the meniscus into air. The vapor transport in the gas domain has a significant effect on the vapor pressure near the meniscus, and hence on local evaporation rates. It is observed from Fig. 6 that the highest vapor concentration in the gas domain is approximately 6.5 mol/m3, and occurs in the meniscus region. Sharper gradients in vapor concentration
Near heated wall Interfacial Temperature (K)
The heat flow paths are illustrated in Fig. 3, and consist of the following:
322
321
320
Near center wall 3.2. Evaporation and vapor transport at the meniscus 319
The interfacial temperature along the meniscus is plotted in Fig. 4. The highest temperature along the interface of 322.8 K occurs at the contact line with the heated wall, while the lowest
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
Horizontal direction to the center wall (m)
2
Evaporation Mass Flux (kg/sm )
Fig. 4. Interface temperature variation along the meniscus.
Near heated wall
0.4
0.3
0.2
0.1
Near center wall 0
0
0.0005
0.001
Horizontal direction to the center wall (m) Fig. 3. Heat flow paths in the V-groove system, along with streamlines and temperature contours (K).
Fig. 5. Local evaporation mass flux variation along the meniscus.
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Fig. 6. Contours of heptane vapor distribution (mol/m3) near the meniscus along with streamlines.
found at both contact lines near the heated wall and the vertical center wall provide higher driving forces for the vapor transport, thus allowing stronger local evaporation. 3.3. Marangoni convection in the liquid It is clear from Fig. 4 that the interfacial temperature along the meniscus is not uniform. This gives rise to Marangoni convection – as illustrated in Fig. 7(a), liquid at the interface is accelerated in the direction of the temperature drop and reaches a maximum value of approximately 0.04 m/s close to the contact line at the center wall. A vortex is thus induced in the liquid which serves to enhance the heat dissipation from the heated V-groove wall. As illustrated in Fig. 7(b), the temperature field in the liquid is also strongly modified by the presence of the Marangoni vortex. The influence of Marangoni convection on the heat transfer rate is explored by considering a case where the surface tension coefficient dr/dT is reduced to 6.2 105 N/(m K) instead of the previous value of 10.26 105 N/(m K). Under these conditions, the maximum interfacial velocity decreases to approximately 0.03 m/s from the earlier value of 0.04 m/s, with a corresponding decrease in evaporation heat transfer to 5.53 W/m from 5.54 W/ m. It is therefore clear that while the temperature field in the liquid is affected by Marangoni convection, the amount of heat transferred is not sensitive to this change. 3.4. Convection in the gas Several factors contribute to inducing convection in the gas domain: generation of vapor from the meniscus, natural convection due to heating, and natural convection due to the density difference between heptane vapor and air. The streamlines in the gas domain were previously illustrated in Fig. 3. The convective flows on the upper and lower sides of the heated groove wall are seen to be different. The reasons for these differences are detailed in Figs. 8 and 9, where the streamlines are shown together with the density distribution, vapor concentration, and velocity magnitudes. In Fig. 9(b) strong vapor flow is observed near the contact line with the heated wall due to the high local evaporation rates, as marked by the dashed circle. On the right side of the groove, the gas near the wall expands due to the higher temperature of the wall as seen in Fig. 8(a), and an upward natural convection flow is induced. On the left side of the vertical wall, however, a strong downward flow is observed.
Fig. 7. Flow and temperature fields in the liquid: (a) Contour of velocity magnitude (m/s) with streamlines and (b) contour of temperature (K).
This is attributed to the higher concentration of the heavier heptane vapor in the gas domain in this region as shown in Fig. 8(b), which causes the observed downward flow. 3.5. Wall temperature The overall temperature distribution in the domain was presented earlier in Fig. 3. The portion of the heated groove wall which is not wetted displays the highest temperatures; the highest temperature in the computation is approximately 334 K, which matches the measured value of 337 K [18] reasonably well. The temperature variation along the top surface of the heated V-groove wall is plotted in Fig. 10. The dry region and the wetted region are separated by the three-phase contact line, at which a sharp change in the temperature distribution is observed. The high-resolution temperature measurements in [18] recorded a temperature near the contact line of approximately 323 K, with a small temperature valley of magnitude on the order of 0.1 K observed near the contact line. A similar temperature valley is also
H. Wang et al. / International Journal of Heat and Mass Transfer 54 (2011) 3015–3023
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Fig. 8. (a) Streamlines in the gas with density contours (kg/m3) and (b) contours of vapor concentration (mol/m3).
observed in the model predictions as shown in Fig. 10, and is attributed to the strong local evaporation that was shown in Fig. 5. The predicted temperature near the contact line also matches the experimental measurements to a reasonable degree. 3.6. Thin film below 1 lm thickness As described in the model formulation, the thin film region with thickness less than 1 lm was not included in the simulation. Indeed the model would be more comprehensive if this omitted thin film region (of thickness below 1 lm) were considered. However modeling this nanoscale thin film is challenging. Since the thin film in our case is open to ambient air rather than being in saturated vapor, this makes the treatment of this region even more complicated than studies in the literature that have considered a saturated vapor domain. To confirm that the heat transfer contribution of the omitted thin film region in the present V-groove system is negligible, a simplified coupling model was formulated. The modeling strategy is illustrated in Fig. 11. A virtual thin film is considered beyond the current truncation past 1 lm thickness at the end of the meniscus. Different values for the length of this added thin film were explored to examine its effect on the overall heat transfer. The calculation procedure of the evaporation flux on the thin film was the same as that for the meniscus, with two assumptions: (1) the thin
Fig. 9. Contours of velocity magnitude (m/s) in the gas: (a) Full domain and (b) details near the meniscus.
film temperature was assumed to be the local solid wall temperature Tw; and (2) the suppression of evaporation due to the disjoining pressure is not taken into account. As a result, the evaporation rate from this added thin film region was 3.9 107 kg/s when its length was assumed to be 5 lm, which is two orders of magnitude lower than the overall evaporation rate calculated with the truncated meniscus, i.e., 1.58 105 kg/s. When the length of the additional thin film region was tripled to 15 lm, the rate of evaporation from the thin film region increased by roughly 1.5 times to 6.3 107 kg/s, which is still very small compared to the overall evaporation rate. It was further estimated that even if the thin film length were further tripled to 45 lm, the evaporation rate would increase to 10 107 kg/s, which is still not a significant contributor to the overall evaporation rate. The reason of this small contribution of the thin film region can be understood as follows. First, the total meniscus length in the V groove system considered is 1.4 mm, while the thin film length is only of the order of lm. In some studies in the literature (e.g., [10]), the nanoscale thin-film region was important because the meniscus length was smaller (<400 lm). Second, the meniscus in
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334 325
Wall Temperature (K)
332 330
324.5
328 324
326 324
323.5
322
wetted 0
0.001
dry 0.002
0.003
Horizontal direction to the center wall (m)
323
0.00122
0.00124
0.00126
0.00128
0.0013
Horizontal direction to the center wall (m)
Fig. 10. Temperature distribution along the top surface of the heated wall, with the inset showing a small temperature valley near the contact line.
Fig. 11. Schematic illustration of analysis of an additional thin film region.
this case is open to ambient air rather than to a saturated vapor domain. The strong evaporation in the thin film region generates and adds vapor to the surrounding gas domain, increasing the vapor pressure near the meniscus and thus suppressing the evaporation, which prevents the overall evaporation rate from increasing significantly. As Fig. 12 shows, the evaporation intensity at the end of the meniscus drops significantly when the thin film region was included. A more comprehensive model including an accurate description of the nanoscale thin film region is targeted in future work by the authors.
Fig. 12. Evaporation mass flux near the meniscus end with and without an additional thin-film region being coupled.
the meniscus accounts for 43% of total heat input to the V-groove. The thin liquid film near contact line with the heated wall displays superior heat transfer characteristics. A small temperature valley is obtained at the contact line, which is again consistent with the experimental observation. The convection in the gas domain is shown to have a significant influence on the heat transfer process. A single Marangoni vortex is induced in the liquid due to the nonuniform temperature along the meniscus interface. Acknowledgments
4. Conclusions A numerical investigation is conducted for an evaporation from a liquid meniscus in a heated V-groove which is open to air. The evaporation is coupled to the vapor transport in the gas domain. The effect of evaporative cooling as well as Marangoni effects at the interface is considered. The natural convection in the gas domain due to vapor generation as well as thermal and vapor concentration gradients is simulated. The calculated evaporation rate is consistent with that obtained in experiments conducted in a prior study [18]. Evaporation from
The authors acknowledge support for this work from the Cooling Technologies Research Center, a National Science Foundation Industry/University Cooperative Research Center at Purdue University. This work was also financially supported by the National Natural Science Foundation of China, Grant No. 50706001. References [1] X. Xu, V.P. Carey, Film evaporation from a micro-grooved surface – an approximate heat transfer model and its comparison with experimental data, J. Thermophys. 4 (1990) 512–520.
H. Wang et al. / International Journal of Heat and Mass Transfer 54 (2011) 3015–3023 [2] D. Khrustalev, A. Faghri, Heat transfer during evaporation on capillary-grooved structures of heat pipes, ASME J. Heat Transfer 117 (1995) 740–747. [3] J.M. Ha, G.P. Peterson, The interline heat transfer of evaporating thin films along a micro-grooved surface, ASME J. Heat Transfer 118 (1996) 747–755. [4] P.C. Stephan, C.A. Busse, Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls, Int. J. Heat Mass Transfer 35 (1992) 383–391. [5] J.P. Burelbach, S.G. Bankoff, S.H. Davis, Nonlinear stability of evaporating/ condensing liquid film, J. Fluid Mech. 195 (1988) 463–494. [6] G.R. Schmidt, Thermocapillary flow about an evaporating meniscus, in: AIAA 27th Thermophysics Conference, Nashville, TN, USA, 1992. [7] B.V. Deryagin, S.V. Nerpin, N.V. Churayev, Effect of film heat transfer upon evaporation of liquids from capillaries, Bull. R.I.L.E.M. 29 (1965) 93–98. [8] P.C. Wayner Jr., Y.K. Kao, L.V. LaCroix, The interline heat transfer coefficient of an evaporating wetting film, Int. J. Heat Mass Transfer 19 (1976) 487–492. [9] C. Chakraborty, S.K. Som, Heat transfer in an evaporating thin liquid film moving slowly along the walls of an inclined microchannel, Int. J. Heat Mass Transfer 48 (2005) 2801–2805. [10] H. Wang, S.V. Garimella, J.Y. Murthy, Characteristics of an evaporating thin film in a microchannel, Int. J. Heat Mass Transfer 50 (2007) 3933–3942. [11] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature 389 (1997) 827–829. [12] M. Cachile, O. Benichou, C. Poulard, A.M. Cazabat, Evaporating droplets, Langmuir 18 (2002) 8070–8078. [13] H.K. Dhavaleswarapu, C.P. Migliaccio, S.V. Garimella, J.Y. Murthy, Experimental investigation of evaporation from low-contact-angle sessile droplets, Langmuir 26 (2010) 880–888. [14] C. Buffone, K. Sefiane, IR measurements of interfacial temperature during phase change in a confined environment, Exp. Therm. Fluid Sci. 29 (2004) 65–74.
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[15] C. Buffone, K. Sefiane, J.R.E. Christy, Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tube using micro-particle image velocimetry, Phys. Fluids 17 (2005) 052104. [16] P. Chamarthy, H.K. Dhavaleswarapu, S.V. Garimella, J.Y. Murthy, S.T. Wereley, Visualization of convection patterns near an evaporating meniscus using lPIV, Exp. Fluids 44 (2008) 431–438. [17] Z. Pan, H. Wang, Symmetry-to-asymmetry transition of Marangoni flow at a convex volatizing meniscus, Microfluidics Nanofluidics 9 (2010) 657–669. [18] C.P. Migliaccio, H.K. Dhavaleswarapu, S.V. Garimella, Temperature measurements near the contact line of an evaporating meniscus in a Vgroove, Int. J. Heat Mass Transfer 54 (2011) 1520–1526. [19] R.W. Schrage, A Theoretical Study of Interface Mass Transfer, Columbia University Press, New York, 1953. [20] A. Faghri, Heat Pipe Science and Technology, Taylor & Francis, Washington, DC, 1995. [21] S.M. Yang, Improvement of the basic correlating equations and transition criteria of natural convection heat transfer, Heat Transfer – Asian Res. 30 (2001) 293–299. [22] D. Juric, G. Tryggvason, Computation of boiling flows, Int. J. Multiphase Flow 24 (1998) 387–410. [23] U. Vadakkan, J.Y. Murthy, S.V. Garimella, Transient analysis of flat heat pipes, in: Proceedings of HT2003, 47349. [24] U. Vadakkan, S.V. Garimella, J.Y. Murthy, Transport in flat heat pipes at high heat fluxes from multiple discrete sources, ASME J. Heat Transfer 126 (2004) 347–354. [25] S.R. Mathur, J.Y. Murthy, A pressure-based method for unstructured meshes, Numer. Heat Transfer 31 (2) (1997) 195–216. [26] J.Y. Murthy, S.R. Mathur, A conservative numerical scheme for the energy equation, ASME J. Heat Transfer 120 (1998) 1081–1085. [27] Fluent Inc., Fluent 6.2 User’s Guide, 2004. [28] NIST Chemistry Webbook, 2009.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical investigation of heat and mass transfer from an evaporating meniscus in a heated open groove Hao Wang a,⇑, Zhenhai Pan a, Suresh V. Garimella b a b
Energy and Resources Engineering Department, College of Engineering, Peking University, Beijing 100871, China Cooling Technologies Research Center, An NSF IUCRC, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA
a r t i c l e
i n f o
Article history: Received 30 July 2010 Received in revised form 29 January 2011 Accepted 18 February 2011 Available online 22 March 2011 Keywords: Evaporation Vapor transport Diffusion Meniscus Thin film Marangoni Groove Temperature valley
a b s t r a c t The process of evaporation from a meniscus into air is more complicated than in enclosed chambers filled with pure vapor. The vapor pressure at the liquid–gas interface depends on both of the evaporation and the vapor transport in the gas environment. Heat and mass transport from an evaporating meniscus in an open heated V-groove is numerically investigated and the results are compared to experiments. The evaporation is coupled to the vapor transport in the gas domain. Conjugate heat transfer is considered in the solid walls, and the liquid and gas domains. The flow induced in the liquid due to Marangoni effects, as well as natural convection in the gas due to thermal expansivity and vapor concentration gradients are simulated. The calculated evaporation rates are found to agree reasonably well with experimentally measured values. The convection in the gas domain has a significant influence on the overall heat transfer and the wall temperature distribution. The evaporation rate near the contact lines on either end of the meniscus is high. Heat transfer through the thin liquid film near the heated wall is found to be very efficient. A small temperature valley is obtained at the contact line which is consistent with the experimental observation. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction A variety of industrial applications have exploited the high heat transfer rates that can be supported by the phase change occurring in evaporating liquid menisci in capillary structures. A detailed understanding of the heat and mass transfer occurring in evaporation from menisci under different operating conditions is critical to the proper design of a variety of passive phase-change devices. A V-shaped groove is a type of capillary structure that has been widely studied. In many of the past studies, a closed system was considered in which the vapor domain was assumed to be uniform at saturated conditions, including in the region near the meniscus. Xu and Carey [1] developed an approximate model for meniscus evaporation in a V-groove. They assumed that the liquid flow along the groove was driven primarily by the capillary pressure difference due to the recession of the meniscus towards the apex of the groove. Khrustalev and Faghri [2] presented a mathematical model for the heat transfer through thin liquid films in the evaporator section of heat pipes with capillary grooves. The model accounted for the effects of interfacial thermal resistance, disjoining pressure, and surface roughness for a given meniscus contact angle. An analytical investigation of the heat transfer characteris⇑ Corresponding author. Tel.: +86 10 82529060. E-mail address: [email protected] (H. Wang). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.02.047
tics of evaporating thin liquid films in V-shaped microgrooves was conducted by Ha and Peterson [3]. The extended meniscus was divided into three regions: absorbed layer or non-evaporating region, evaporating thin film or transition region, and intrinsic meniscus region. Stephan and Busse [4] calculated the heat and mass transfer in the micro-region and then combined the solution with a treatment of the macroscopic meniscus within open grooves. Burelbach et al. [5] conducted a theoretical analysis of evaporating/condensing liquid films, in which vapor recoil and Marangoni convection were included. Schmidt [6] discussed the influence of Marangoni and buoyancy effects on the flow field near an evaporating meniscus. A number of studies [7–10] have also investigated the disjoining pressure due to long-range molecular forces as well as evaporation from the thin liquid film at the contact line. The transport process is more complicated, however, if the gas domain is not uniformly saturated with vapor but contains noncondensable gases as well. Although evaporation into unsaturated vapor domains is encountered widely in applications, it is less well-understood. Deegan et al. [11] investigated the drying process of water droplets in open space, showing that the evaporation from droplets was influenced by diffusion of vapor in the air. Cachile et al. [12] also investigated droplet drying in air and explained their experimental results by considering vapor transport in air. Dhavaleswarapu et al. [13] used micro-particle image velocimetry
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Nomenclature A C D Gr J hfg k l m mnet m00net M M n p R R Re T t V
area (m2) vapor molar concentration (mol/m3) diffusion coefficient in air (m2/s) Grashof number molar flux (mol /(m2 s)) latent heat of evaporation (J/kg) thermal conductivity (W/(m K)) characteristic length (m) mass (kg) mass flow rate (kg/s) mass flux (kg/(m2 s)) molecular weight (kg/mol) molecular weight (kg/kmol) interface normal coordinate (m) pressure (N/m2) radius (m) universal gas constant (J/(mol K)) Reynolds number temperature (K) time (s) fluid velocity (m/s)
r s
thickness (m) contact angle kinematic viscosity (m2/s) dynamic viscosity ((N s)/m2) density (kg/m3) surface tension coefficient (N/m) interface accommodation coefficient shear stress (N/m2)
Subscript diff e equ g l lv out ref sat v
diffusion evaporation equilibrium gas (vapor/air mixture) liquid interface outlet reference saturated vapor
d h
m l q r
Greek symbols b thermal expansion coefficient (1/K)
to reconstruct velocity fields in sessile water droplets evaporating into air. The spatial and temporal distribution of the local evaporative mass flux was determined, and the temperature distribution in the droplet near the contact line was estimated. Buffone et al. [14,15] conducted experiments on a meniscus in a microtube which was open to air. Similar experiments in [16] delineated Marangoni convection in capillary tubes. More recently, Pan and Wang [17] numerically investigated the broken symmetry of the Marangoni convections in micro-tubes, where a non-uniform vapor diffusion could be with responsibility. Migliaccio et al. [18] recently conducted detailed experiments for an evaporating heptane meniscus in a V-groove open to air. The groove was made of fused quartz, and electrical heating was provided by a thin layer of titanium coated on the backside of the quartz substrate. The effects of liquid feeding rate on the temperature suppression and meniscus shape were explored. Highresolution infrared thermography was employed to investigate the temperature profile on the groove wall. A small temperature valley was recorded at the contact line, indicating the localized cooling due to thin film heat transfer. The present work reports on a numerical investigation conducted for the evaporating meniscus in the open V-groove under conditions matching those of the experiments in [18]. The evaporation and vapor transport are coupled at the meniscus in the numerical model. The evaporation process, cooling effect and conduction in the groove wall, Marangoni flow, and convection in the vapor are all comprehensively investigated. The predicted results are compared to those from the experiments. 2. Mathematical model 2.1. Problem description The experiments in [18] considered a V-groove setup consisting of an angled heated quartz wall and a vertical center wall as shown in Fig. 1, simulating one half of a V-groove. The outer surface of the quartz wall is heated by means of electrical power supplied to a thin metallic film deposited on this surface. A meniscus is
sustained between the two walls, and its shape is determined by optical means. The half V-groove in the experiments is 50 mm long in the axial (z-) direction. To replace the liquid lost to evaporation, a syringe pump continuously feeds room-temperature liquid to the groove through a capillary tube positioned just above the meniscus contact line, 12.5 mm from one end of the groove. For the present modeling work, the liquid meniscus profile obtained at a liquid supply rate of 6.55 107 kg/s and an applied heat flux of 1800 W/m2 is chosen for investigation. A two-dimensional representation of the problem is adopted as shown in Fig. 1, for computational tractability. The mass loss due to evaporation in the liquid is not considered in the model since there is no liquid supply inlet in the 2D representation; this is a reasonable simulation of the experiment since the liquid consumption in the experiment is of order of 1 105 kg/(s m), with corresponding induced liquid flow velocities of less than 1 105 m/s. 2.2. Evaporation and vapor transport at the meniscus The evaporation from a liquid–gas meniscus is determined by several factors. The interfacial temperature Tlv is important since a higher Tlv brings more higher-energy liquid molecules to cross the interface into the vapor phase. The vapor pressure near the interface pv_lv is also an important factor because a higher pv_lv means that more vapor molecules could cross the interface back into the liquid phase, thus reducing the net evaporation mass flux m00net ; evaporation into a vacuum is more intensive than that into a vapor domain. When evaporating into air, the vapor molecules must diffuse through the air under a driving force. Thus, the vapor pressure near the interface pv_lv must be higher than that far from the interface. To be complete, analysis of evaporation at the meniscus must account for vapor transport in the gas domain. The net evaporated mass flux across the interface may be calculated according to the molecular kinetics-based evaporation theory of Schrage [19]:
M00net
2r M ¼ 2 r 2pR
!1=2
pv
equ ðT lv Þ 1=2 T lv
p v1=2lv T v lv
! ð1Þ
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With appropriate boundary conditions, the distribution of vapor concentration in the gas domain can be computed, and the mass flux due to vapor transport at the liquid–gas interface can be obtained as
Gravity Groove wall: quartz
0.004 0.003
Left side Heated underside
0.002
y (m)
m00v ¼ M J ¼ M ðD
0.001
Vapor / Air
0.000
Liquid
Meniscus in experiment
v n j lv ¼ Sealant junction
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007
x (m)
m00v ¼
C total ¼ 0.060
Boundary
MD @C v 1 C v =C total @n lv
ð7Þ
p RT lv
ð8Þ
and the vapor mole concentration Cv at the interface is determined from pv_lv under the ideal gas assumption:
Vapor / Air
0.040
p C v jlv ¼ v lv C total p
0.020
The liquid mass flux evaporated from the interface must be equal to the vapor mass flux at the interface by mass conservation, i.e., m00net ¼ m00v , which couples the evaporation rate to vapor transport at the interface. Combining Eqs. (3) and (7),
0.000
ð9Þ
Impermeable wall
pv lv ¼ pv
-0.020 -0.05
0
in which pv_equ(Tlv) is the equilibrium vapor pressure at which there would be no net mass flux across the interface. It is approximated as the corresponding saturation pressure, despite the implicit omission of suppression effects due to the disjoining pressure and the capillary pressure [20] in this approximation. The saturation pressure psat(Tlv) is calculated by the well-known Clausius–Clapeyron equation: equ ðT lv Þ psat ðT lv Þ ¼ psat
ref
exp
Mhfg R
1
T sat
ref
1 T lv
! ð2Þ
Assuming that the vapor temperature at the interface Tv_lv is equal to the liquid temperature at the interface Tlv,
m00net ¼
2r M 2 r 2pR
!1=2
1 T l1=2 v
ðpv
equ ðT lv Þ
pv lv Þ
ð3Þ
When the evaporation occurs into a uniformly vapor-saturated domain, pv_lv in Eq. (3) is known, and the evaporation mass flux m00net is obtained. In the present work, where the evaporation takes place in an air domain, pv_lv is unknown but may be calculated from a solution of the vapor transport problem. The governing equation for vapor transport in air, including the diffusion and convection components, is given as
@C v v rC v ¼ r ðD rC v Þ þ~ @t
equ ðT lv Þ
0.05
x (m)
Fig. 1. Schematic diagram of the open V-groove with an evaporating meniscus in the experiments [18].
pv
ð6Þ
where the total gas mole concentration Ctotal is
0.080
y (m)
1 @C air 1 @C v D ¼ D C air C total C v @n lv @n lv
Substituting this expression for vn into Eq. (5), the mass flux at the interface due to vapor transport is
Central wall: glass
-0.003 -0.001
ð5Þ
The first term on the right is the diffusion component while the second is the convection component; the use of vn ensures that the net mass transport of air is zero since air does not cross the interface, based on which the following expression may be written:
-0.001 -0.002
@C v þ v n C v Þ @n lv
ð4Þ
2r 2r
m00v 1=2
M 2pR
ð10Þ 1 1=2 T lv
In the computations, the vapor distribution Cv in the gas domain is first calculated with the initial boundary conditions. The vapor mass flux at the interface m00v is then obtained from Eq. (7). Eq. (10) is then used to calculate pv_lv, after which Cv at the interface is updated for the next round of vapor transport calculations. 2.3. Gas and liquid domains As explained above, vapor transport in the gas domain strongly influences evaporation at the meniscus. Careful calculation of the flow, temperature, and vapor concentration fields is therefore necessary. In the present work, the flow in the gas domain is assumed to be laminar and Newtonian, and the ideal-gas law is assumed to hold. With g = 9.8 m/s2, m = 1.6 105 m2/s, b = 0.0031 K1, and approximating l 0.01 m and DT 40 K, a value of the Grashof number (Gr = gl3bDT/m2) in the gas domain of Gr 4.7 103 is estimated, which justifies the assumption of laminar flow [21]. The density of the gas–vapor mixture (heptane vapor is heavier than air) is given by:
q ¼ C v Mv þ
p TR
C v M air
ð11Þ
and is not only a function of the local temperature T, but also the local concentration Cv. The continuity, momentum and energy equations must be solved together with the vapor transport equations (Eqs. (4) and (11)) to provide a complete description of convection in the gas domain. Boundary conditions on Cv are set as follows: the value of Cv at the interface is determined by Eq. (10), while the value at the outer boundary of the gas domain (a circle which is 0.1 m in diameter as
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shown in Fig. 1, 14 times the groove length) is assumed to be zero. The adequacy of the boundary size for the simulation is verified by considering a domain diameter that is half the selected size. In both cases, the evaporation mass flow rate was found to be 1.58 105 kg/(s m). The far-field boundary condition (Cv = 0) is also verified by running the simulation with a larger value of
Cv = 0.2 mol/m3. This only changes the evaporation mass flow rate very slightly to 1.57 105 kg/(s m). Experimental measurements of Cv in the V-groove domain would allow for greater fidelity in the numerical model. The flow in the liquid domain is assumed to be laminar, incompressible and Newtonian. The Boussinesq approximation is employed for modeling natural convection in the liquid, whereby the density changes with temperature only in the buoyancy term. Marangoni effects are also included. Mass transport across the interface is implemented in the layer of mesh cells adjacent to the interface on either side, as explained in [22–24]. The increase in mass in the gas due to evaporation is taken into account. A negative energy source term is employed on the liquid side to simulate the cooling effect due to evaporation. 2.4. Numerical analysis The numerical solution is obtained using the pressure-based finite volume scheme described in [25,26], and implemented in the commercial software package, FLUENT [27]. Pressure–velocity coupling is accomplished through the SIMPLE algorithm. The outer boundary is designated as a pressure-outlet boundary. The domain is discretized into quadrilateral elements as shown in Fig. 2. The very thin liquid film near the contact line is truncated in the mesh so that a film thickness less than 1 lm is neglected. 3. Results and discussion Results from the numerical model are presented for heptane evaporating in a V-groove made of quartz for the heated wall and glass for the center wall, to simulate the experiments in [18]. The parameters used in the simulation are listed in Tables 1 and 2. The predicted results are compared to the experimental measurements from [18]. 3.1. Heat transfer analysis In the experiment [18] the heat flux imposed on the quartz wall was 1800 W/m2. The wall width and length were 7.1 mm and 50 mm, respectively. The total heat imposed on the groove was therefore 0.64 W. For the present 2D model, this translates to a heat input of 12.78 W/m to simulate experimental conditions. Table 1 Fluid properties [28]. Thermophysical properties
Heptane
Air
Density (kg/m3)
666.95 at 294.65 K
Thermal conductivity (W/m K) Heat capacity (J/kg) Viscosity (kg/m s) Vapor molecular weight Thermal expansion coefficient dr/dT (N/m K) (313 K to 323 K) Vapor diffusion coefficient in air (m2/s) Accommodation coefficient Latent heat of evaporation (J/kg) Saturated vapor pressure (Pa)
0.12746 2298.4 0.000331 100 0.00128 0.000103 7.2 106 at 294.65 K 1 3.506 105 5402 at 294.65 K
Defined in Eq. (11) 0.0242 1006.43 1.789 105 29 – – – – – –
Table 2 Solid properties [18].
Fig. 2. Mesh setup for the V-groove system: (a) the full mesh, (b) details of the mesh in the vicinity of the groove, and (c) mesh near the contact line (the liquid film thinner than 1 lm is neglected).
Thermophysical properties
Fused quartz
Glass
RTV sealant
Density (kg/m3) Thermal conductivity (W/m K) Heat capacity (J/kg)
2200 1.4 670
2500 1.1 840
1500 0.2 920
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a. The evaporation heat transfer, Qe, is calculated from the model to be 5.54 W/m, accounting for 43.3% of the total heat input. The corresponding evaporation mass flow rate is 1.58 105 kg/(s m). This value is comparable to the liquid supply rate in the experiments of approximately 1.31 105 kg/(s m). b. The convection heat transfer from the unwetted portions of the groove wall, Qc1, contributes 4.66 W/m, accounting for 36.5% of the total heat input. This includes convection from the underside (2.53 W/m), the unwetted portion of the top side (1.62 W/m), the top edge (0.34 W/m), and the sealant area (0.17 W/m). c. The convection heat transfer from the vertical center wall, Qc2, is 1.28 W/m, which is 10% of the total input. d. In the experiments, the evaporating liquid is at the local temperature of the interface, while the liquid supply to the groove is at the room temperature of approximately 294.65 K. Therefore some of the heat input is absorbed by the supplied liquid to heat up to the interfacial temperature. This component Qs is directly calculated based on the simulated evaporation rate and the difference between room temperature and interface temperature to be 0.96 W/m, which is 7.5% of the total input. Since the liquid mass loss at the interface is not simulated in the model, this component of heat transfer is accounted for by subtracting it from the total heat input, effectively decreasing the applied heat flux at the heated boundary to 1665 W/m2 from 1800 W/m2. e. A final component Qc3 is dissipated from the meniscus by convection rather than by evaporation, and amounts to 0.35 W/m; this accounts for 2.7% of the total heat input, and is thus negligible. Of the total heat transfer, Qe + Qc1 + Qc2 + Qc3 + Qs = 12.78 W/m, phase change accounts for 43.3% while the convective components account for more than 50% of the total heat input.
value of 319 K is observed at the contact line with the vertical center wall. The local evaporation mass flux along the meniscus is plotted in Fig. 5. Near the contact line at the heated wall, the strongest evaporation flux is observed with a maximum of approximately 0.43 kg/(s m2), corresponding to a local heat flux of 1.3 105 W/ m2. This strong evaporation is due to the high local interfacial temperature. While the evaporation rate drops rapidly with distance away from the heated-wall contact line, there is a small increase as the contact line with the vertical center wall is approached. While this region has the lowest interfacial temperature, the improvement in evaporation rate is likely caused by the ease of vapor dissipation in this region, as can be surmised from the Cv contours shown in Fig. 6. The advantageous geometry in the region near the center-wall contact line facilitates vapor diffusion from the meniscus into air. The vapor transport in the gas domain has a significant effect on the vapor pressure near the meniscus, and hence on local evaporation rates. It is observed from Fig. 6 that the highest vapor concentration in the gas domain is approximately 6.5 mol/m3, and occurs in the meniscus region. Sharper gradients in vapor concentration
Near heated wall Interfacial Temperature (K)
The heat flow paths are illustrated in Fig. 3, and consist of the following:
322
321
320
Near center wall 3.2. Evaporation and vapor transport at the meniscus 319
The interfacial temperature along the meniscus is plotted in Fig. 4. The highest temperature along the interface of 322.8 K occurs at the contact line with the heated wall, while the lowest
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
Horizontal direction to the center wall (m)
2
Evaporation Mass Flux (kg/sm )
Fig. 4. Interface temperature variation along the meniscus.
Near heated wall
0.4
0.3
0.2
0.1
Near center wall 0
0
0.0005
0.001
Horizontal direction to the center wall (m) Fig. 3. Heat flow paths in the V-groove system, along with streamlines and temperature contours (K).
Fig. 5. Local evaporation mass flux variation along the meniscus.
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Fig. 6. Contours of heptane vapor distribution (mol/m3) near the meniscus along with streamlines.
found at both contact lines near the heated wall and the vertical center wall provide higher driving forces for the vapor transport, thus allowing stronger local evaporation. 3.3. Marangoni convection in the liquid It is clear from Fig. 4 that the interfacial temperature along the meniscus is not uniform. This gives rise to Marangoni convection – as illustrated in Fig. 7(a), liquid at the interface is accelerated in the direction of the temperature drop and reaches a maximum value of approximately 0.04 m/s close to the contact line at the center wall. A vortex is thus induced in the liquid which serves to enhance the heat dissipation from the heated V-groove wall. As illustrated in Fig. 7(b), the temperature field in the liquid is also strongly modified by the presence of the Marangoni vortex. The influence of Marangoni convection on the heat transfer rate is explored by considering a case where the surface tension coefficient dr/dT is reduced to 6.2 105 N/(m K) instead of the previous value of 10.26 105 N/(m K). Under these conditions, the maximum interfacial velocity decreases to approximately 0.03 m/s from the earlier value of 0.04 m/s, with a corresponding decrease in evaporation heat transfer to 5.53 W/m from 5.54 W/ m. It is therefore clear that while the temperature field in the liquid is affected by Marangoni convection, the amount of heat transferred is not sensitive to this change. 3.4. Convection in the gas Several factors contribute to inducing convection in the gas domain: generation of vapor from the meniscus, natural convection due to heating, and natural convection due to the density difference between heptane vapor and air. The streamlines in the gas domain were previously illustrated in Fig. 3. The convective flows on the upper and lower sides of the heated groove wall are seen to be different. The reasons for these differences are detailed in Figs. 8 and 9, where the streamlines are shown together with the density distribution, vapor concentration, and velocity magnitudes. In Fig. 9(b) strong vapor flow is observed near the contact line with the heated wall due to the high local evaporation rates, as marked by the dashed circle. On the right side of the groove, the gas near the wall expands due to the higher temperature of the wall as seen in Fig. 8(a), and an upward natural convection flow is induced. On the left side of the vertical wall, however, a strong downward flow is observed.
Fig. 7. Flow and temperature fields in the liquid: (a) Contour of velocity magnitude (m/s) with streamlines and (b) contour of temperature (K).
This is attributed to the higher concentration of the heavier heptane vapor in the gas domain in this region as shown in Fig. 8(b), which causes the observed downward flow. 3.5. Wall temperature The overall temperature distribution in the domain was presented earlier in Fig. 3. The portion of the heated groove wall which is not wetted displays the highest temperatures; the highest temperature in the computation is approximately 334 K, which matches the measured value of 337 K [18] reasonably well. The temperature variation along the top surface of the heated V-groove wall is plotted in Fig. 10. The dry region and the wetted region are separated by the three-phase contact line, at which a sharp change in the temperature distribution is observed. The high-resolution temperature measurements in [18] recorded a temperature near the contact line of approximately 323 K, with a small temperature valley of magnitude on the order of 0.1 K observed near the contact line. A similar temperature valley is also
H. Wang et al. / International Journal of Heat and Mass Transfer 54 (2011) 3015–3023
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Fig. 8. (a) Streamlines in the gas with density contours (kg/m3) and (b) contours of vapor concentration (mol/m3).
observed in the model predictions as shown in Fig. 10, and is attributed to the strong local evaporation that was shown in Fig. 5. The predicted temperature near the contact line also matches the experimental measurements to a reasonable degree. 3.6. Thin film below 1 lm thickness As described in the model formulation, the thin film region with thickness less than 1 lm was not included in the simulation. Indeed the model would be more comprehensive if this omitted thin film region (of thickness below 1 lm) were considered. However modeling this nanoscale thin film is challenging. Since the thin film in our case is open to ambient air rather than being in saturated vapor, this makes the treatment of this region even more complicated than studies in the literature that have considered a saturated vapor domain. To confirm that the heat transfer contribution of the omitted thin film region in the present V-groove system is negligible, a simplified coupling model was formulated. The modeling strategy is illustrated in Fig. 11. A virtual thin film is considered beyond the current truncation past 1 lm thickness at the end of the meniscus. Different values for the length of this added thin film were explored to examine its effect on the overall heat transfer. The calculation procedure of the evaporation flux on the thin film was the same as that for the meniscus, with two assumptions: (1) the thin
Fig. 9. Contours of velocity magnitude (m/s) in the gas: (a) Full domain and (b) details near the meniscus.
film temperature was assumed to be the local solid wall temperature Tw; and (2) the suppression of evaporation due to the disjoining pressure is not taken into account. As a result, the evaporation rate from this added thin film region was 3.9 107 kg/s when its length was assumed to be 5 lm, which is two orders of magnitude lower than the overall evaporation rate calculated with the truncated meniscus, i.e., 1.58 105 kg/s. When the length of the additional thin film region was tripled to 15 lm, the rate of evaporation from the thin film region increased by roughly 1.5 times to 6.3 107 kg/s, which is still very small compared to the overall evaporation rate. It was further estimated that even if the thin film length were further tripled to 45 lm, the evaporation rate would increase to 10 107 kg/s, which is still not a significant contributor to the overall evaporation rate. The reason of this small contribution of the thin film region can be understood as follows. First, the total meniscus length in the V groove system considered is 1.4 mm, while the thin film length is only of the order of lm. In some studies in the literature (e.g., [10]), the nanoscale thin-film region was important because the meniscus length was smaller (<400 lm). Second, the meniscus in
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334 325
Wall Temperature (K)
332 330
324.5
328 324
326 324
323.5
322
wetted 0
0.001
dry 0.002
0.003
Horizontal direction to the center wall (m)
323
0.00122
0.00124
0.00126
0.00128
0.0013
Horizontal direction to the center wall (m)
Fig. 10. Temperature distribution along the top surface of the heated wall, with the inset showing a small temperature valley near the contact line.
Fig. 11. Schematic illustration of analysis of an additional thin film region.
this case is open to ambient air rather than to a saturated vapor domain. The strong evaporation in the thin film region generates and adds vapor to the surrounding gas domain, increasing the vapor pressure near the meniscus and thus suppressing the evaporation, which prevents the overall evaporation rate from increasing significantly. As Fig. 12 shows, the evaporation intensity at the end of the meniscus drops significantly when the thin film region was included. A more comprehensive model including an accurate description of the nanoscale thin film region is targeted in future work by the authors.
Fig. 12. Evaporation mass flux near the meniscus end with and without an additional thin-film region being coupled.
the meniscus accounts for 43% of total heat input to the V-groove. The thin liquid film near contact line with the heated wall displays superior heat transfer characteristics. A small temperature valley is obtained at the contact line, which is again consistent with the experimental observation. The convection in the gas domain is shown to have a significant influence on the heat transfer process. A single Marangoni vortex is induced in the liquid due to the nonuniform temperature along the meniscus interface. Acknowledgments
4. Conclusions A numerical investigation is conducted for an evaporation from a liquid meniscus in a heated V-groove which is open to air. The evaporation is coupled to the vapor transport in the gas domain. The effect of evaporative cooling as well as Marangoni effects at the interface is considered. The natural convection in the gas domain due to vapor generation as well as thermal and vapor concentration gradients is simulated. The calculated evaporation rate is consistent with that obtained in experiments conducted in a prior study [18]. Evaporation from
The authors acknowledge support for this work from the Cooling Technologies Research Center, a National Science Foundation Industry/University Cooperative Research Center at Purdue University. This work was also financially supported by the National Natural Science Foundation of China, Grant No. 50706001. References [1] X. Xu, V.P. Carey, Film evaporation from a micro-grooved surface – an approximate heat transfer model and its comparison with experimental data, J. Thermophys. 4 (1990) 512–520.
H. Wang et al. / International Journal of Heat and Mass Transfer 54 (2011) 3015–3023 [2] D. Khrustalev, A. Faghri, Heat transfer during evaporation on capillary-grooved structures of heat pipes, ASME J. Heat Transfer 117 (1995) 740–747. [3] J.M. Ha, G.P. Peterson, The interline heat transfer of evaporating thin films along a micro-grooved surface, ASME J. Heat Transfer 118 (1996) 747–755. [4] P.C. Stephan, C.A. Busse, Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls, Int. J. Heat Mass Transfer 35 (1992) 383–391. [5] J.P. Burelbach, S.G. Bankoff, S.H. Davis, Nonlinear stability of evaporating/ condensing liquid film, J. Fluid Mech. 195 (1988) 463–494. [6] G.R. Schmidt, Thermocapillary flow about an evaporating meniscus, in: AIAA 27th Thermophysics Conference, Nashville, TN, USA, 1992. [7] B.V. Deryagin, S.V. Nerpin, N.V. Churayev, Effect of film heat transfer upon evaporation of liquids from capillaries, Bull. R.I.L.E.M. 29 (1965) 93–98. [8] P.C. Wayner Jr., Y.K. Kao, L.V. LaCroix, The interline heat transfer coefficient of an evaporating wetting film, Int. J. Heat Mass Transfer 19 (1976) 487–492. [9] C. Chakraborty, S.K. Som, Heat transfer in an evaporating thin liquid film moving slowly along the walls of an inclined microchannel, Int. J. Heat Mass Transfer 48 (2005) 2801–2805. [10] H. Wang, S.V. Garimella, J.Y. Murthy, Characteristics of an evaporating thin film in a microchannel, Int. J. Heat Mass Transfer 50 (2007) 3933–3942. [11] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature 389 (1997) 827–829. [12] M. Cachile, O. Benichou, C. Poulard, A.M. Cazabat, Evaporating droplets, Langmuir 18 (2002) 8070–8078. [13] H.K. Dhavaleswarapu, C.P. Migliaccio, S.V. Garimella, J.Y. Murthy, Experimental investigation of evaporation from low-contact-angle sessile droplets, Langmuir 26 (2010) 880–888. [14] C. Buffone, K. Sefiane, IR measurements of interfacial temperature during phase change in a confined environment, Exp. Therm. Fluid Sci. 29 (2004) 65–74.
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[15] C. Buffone, K. Sefiane, J.R.E. Christy, Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tube using micro-particle image velocimetry, Phys. Fluids 17 (2005) 052104. [16] P. Chamarthy, H.K. Dhavaleswarapu, S.V. Garimella, J.Y. Murthy, S.T. Wereley, Visualization of convection patterns near an evaporating meniscus using lPIV, Exp. Fluids 44 (2008) 431–438. [17] Z. Pan, H. Wang, Symmetry-to-asymmetry transition of Marangoni flow at a convex volatizing meniscus, Microfluidics Nanofluidics 9 (2010) 657–669. [18] C.P. Migliaccio, H.K. Dhavaleswarapu, S.V. Garimella, Temperature measurements near the contact line of an evaporating meniscus in a Vgroove, Int. J. Heat Mass Transfer 54 (2011) 1520–1526. [19] R.W. Schrage, A Theoretical Study of Interface Mass Transfer, Columbia University Press, New York, 1953. [20] A. Faghri, Heat Pipe Science and Technology, Taylor & Francis, Washington, DC, 1995. [21] S.M. Yang, Improvement of the basic correlating equations and transition criteria of natural convection heat transfer, Heat Transfer – Asian Res. 30 (2001) 293–299. [22] D. Juric, G. Tryggvason, Computation of boiling flows, Int. J. Multiphase Flow 24 (1998) 387–410. [23] U. Vadakkan, J.Y. Murthy, S.V. Garimella, Transient analysis of flat heat pipes, in: Proceedings of HT2003, 47349. [24] U. Vadakkan, S.V. Garimella, J.Y. Murthy, Transport in flat heat pipes at high heat fluxes from multiple discrete sources, ASME J. Heat Transfer 126 (2004) 347–354. [25] S.R. Mathur, J.Y. Murthy, A pressure-based method for unstructured meshes, Numer. Heat Transfer 31 (2) (1997) 195–216. [26] J.Y. Murthy, S.R. Mathur, A conservative numerical scheme for the energy equation, ASME J. Heat Transfer 120 (1998) 1081–1085. [27] Fluent Inc., Fluent 6.2 User’s Guide, 2004. [28] NIST Chemistry Webbook, 2009.